Within the conventional neoclassical framework, a distinction is
sometimes made between product-augmenting and factor-augmenting
technical change. A parallel distinction is commonly made between
embodied and disembodied technical change with the former associated
with factor, and the latter with product, augmentation. Disembodied
change is commonly assumed to arise from increases in the stock of
knowledge, independently of the characteristics of the inputs used,
while embodied change relates to increases in the efficiency of inputs,
that is, labor skills or the productivity of physical capital.

Unfortunately, this distinction is ambiguous. Changes in the
efficiency of the inputs used are usually accompanied - indeed made
possible - by increases in knowledge. And conversely, increases in the
stock of knowledge often favor some inputs more than others, including
the capital goods of one vintage relative to those of another.

Notwithstanding the ambiguity, the concept of embodiment has
intuitive appeal and this partly explains the focus on decomposing the
sources of technical change that followed Solow's [6] seminal paper. But by the late 1960s, a reader of the literature might have
concluded that such decomposition was impossible. For with merely time
series data on inputs and output, product-augmenting and
factor-augmenting technical change are empirically indistinguishable.

In an important paper, Hall [3] showed that with data on used
equipment prices and the interest rate, embodied technical change and
the deterioration function can, in principle, be calculated. However,
the paucity of data on the price of used capital goods has allowed
little progress in this direction. The new Longitudinal Research
Database created by the U.S. Bureau of the Census now permits still
another approach to the estimation of "embodied" technical
change associated with capital, both physical and human. In addition, it
casts light on a perplexing problem that has plagued econometric estimates of production relations based on changes in inputs and output
as distinct from levels of both.

This new body of information consists of time series and
cross-section data for individual manufacturing plants for the period
1972 to 1986. The time series permit us to derive indexes of the vintage
of capital for each plant. This, in turn, allows us to estimate the
effects of vintage of capital on productivity from strictly
cross-section data. And since these effects are estimated at a common
point in time, temporal shifts in productivity divorced from vintage are
excluded by definition.

Moreover, we are also able to distinguish between "new"
plants - that is, plants without endowments of capital accumulated in
earlier periods - from "old" plants. The analysis of data for
old plants allows the test of a hypothesis, and yields an explanation,
of why estimates based on changes in inputs and output generally lead to
very different coefficients from those based on levels of the variables.
The latter is an issue with important policy implications given that
most investment in developed economies takes the form of expansion -
that is, changes in inputs - for existing (old) plants.

The remainder of the paper is divided into six sections. In section
II we present our principal model and the definitions of variables in
our production function. Section III reports the estimates for technical
change in the context of levels of inputs and output for new plants.
Section IV discusses the implications of measuring production relations
for changes in inputs and output as distinct from levels while section V
presents estimates for old plants based on changes in the relevant
variables. Section VI compares the results for new and old plants while
section VII is a brief summary of principal conclusions.

II. Model for Measuring Technical Advance and Definition of
Variables

We start with a general model

[Mathematical Expressions Omitted]

Where [O.sub.[tau]] is output, [A.sub.[tau]] is a shift parameter
that is assumed to affect the productivity of all vintages of capital
and all labor skills symmetrically, [e.sup.a[tau]] is disembodied
technical change at rate a, [L.sup.[tau]] is labor, [Q.sup.[tau]] is
human capital, and [SYMBOL NOT CONVERTED] is a vector of investment
streams.

[Mathematical Expressions Omitted]

Where [tau] is the vintage year in which investment is measured,
and [gamma] is the age of the plant.

As is commonly hypothesized, we expect each successive vintage of
investment to be more productive than the last so that,

[Mathematical Expressions Omitted]

We next assume a standard production function approach of
substituting an aggregate capital stock variable for the vector of
investments and take due account of the effect of embodiment by
measuring the average vintage of the stock. Accordingly, we have

O = [Ae.sup.a[tau]] F [(K.sub.v.e.sup.kv], L, Q) (4)

Where K is the sum of investments of various vintages, the
subscript v is the weighted average vintage of the stock with weights
based on the investment of each vintage relative to K, and [e.sup.kv] is
an index of productivity enhancement at constant rate k from
"embodied" effects of vintage (the subscripts of [tau] are
omitted). The capital stock term of the production function,
[K.sub.v.e.sup.kv], is thus converted into efficiency units based on
average vintage.

The resulting model differs in several respects from production
functions that are commonly estimated. First, human capital (labor
skills) enters as a separate argument in the production function rather
than as an adjustment to the measure of labor input. Second, capital is
composed of gross investment streams, rather than net investment, so
that the effect of vintage (that is, obsolescence plus decay) is
estimated within the framework of the model. In this respect, our
approach accords with that of Prucha and Nadiri [5], though we do not
follow them in their assumption that the depreciation rate is
endogenous. It contrasts with the conventional method of inferring
obsolescence and deterioration from assumed economic lives and decay
functions.

Our conceptual framework makes no distinction between the
accumulation of knowledge and changes in the physical attributes of
capital associated with vintage as long as new knowledge is uniquely
related to vintage. Similarly, if new knowledge is uniquely related to
labor skills, no distinction is made between the two. Changes in the
shift parameter, [A.sub.[tau], disappear within a crosssection framework
and only interplant variations in "disembodied" technical
change remain.

Differences across plants in blueprint technology, and in the
knowledge associated with it, are almost certainly uniquely related to
either labor skills or the vintage of physical capital. What, then, is
there left of disembodiment in the context of a cross-section model? It
appears that only the effects of organizational capital, largely in the
form of firm-specific information, remain unaccounted.

We next turn to a more detailed discussion of the variables in
equation (4).

Physical Capital and Vintage

The stock of capital in equation (4) is the sum of deflated gross
investments from the year following the birth of the plant to the year
in which output is measured.(1) Obsolescence is then measured directly
via the production function through estimates of the effect of vintage
on output.

The effects of vintage arise from obsolescence + (physical decay -
maintenance outlays). If, however, as is plausible, maintenance outlays
roughly offset the effects of physical decay at least on current
production (if not also on earnings), the principal source of difference
in the relative efficiency of capital of different vintages is
obsolescence. The implied depreciation rate then, correctly measured,
becomes roughly the dual of capital augmenting technical change.

The foregoing indicates that the assumptions necessary to construct
a net capital stock require implicitly a measure of embodied technical
change of capital. And if physical decay roughly equals maintenance
outlays, then obsolescence is all that needs to be measured to transform
gross into net stocks.

Vintage was measured as the weighted average of the years of the
investment stream for each plant, with weights based on the ratio of the
annual investment for each plant to its total investment over the
relevant period. By definition, a higher average indicated more recent
vintage. Thus vintage measured (inversely) the average age of physical
assets.

Since the productivity of an asset has a lower bound of zero, in
principle, only non-retired assets should be included in the computation
of average vintage. Otherwise, a systematic relation between the stock
of retired assets and average vintage might lead, in the context of a
production function, to distortions in the coefficients of both physical
capital and vintage. However, since the period over which average
vintage was computed was limited to 1973-86, retirements of assets from
the relevant investment streams (as our tests showed) were not large
enough to distort the estimates significantly and were, therefore,
ignored.

Excluded from the model is circulating capital (that is,
inventories). This is justified since inventory accumulation is, at
least partly, unintended and is also a function of expected future
rather than merely current output.

Labor and Human Capital

Our labor variable was intended to approximate pure labor
independently of human capital (labor quality) and was thus measured by
the number of employees for each plant. The index for the average amount
of human capital associated with the labor input was simply the average
wage rate for each plant.(2) In effect, we assumed that all plants have
equal access to the labor market and that differences in average wages
must reflect differences in human capital rather than variations in wage
rates for identical classes of labor.

Our chosen measure is implicitly based on a definition of human
capital as any attribute of labor that increases its productivity.
Across plants at a point in time, we assume that average wage rates
reflect primarily differences in the composition of the work force with
respect to what Becker [1] has called general (as distinct from firm
specific) human capital - that is, human capital the returns to which
are probably captured by the employee. Hence, ad hoc measures of labor
quality based on education, occupation, or demographic attributes are
rendered unnecessary.

Differences in average wage rates across plants in the same
industry were far too large to permit a conclusion that they reflected
regional variations in wages. More specifically, for most industries,
the highest average wage for any plant was roughly three times the
lowest, and the standard deviations were typically between 20 and 30
percent of the mean wage. This represents a far greater variation than
can plausibly be attributed to such factors as unionization, historical
peculiarities, or regional differences in wages.

Even more decisive, if historical accident or unionization were
important explanatory variables for the dispersion in average wages
across plants, one would expect the variation to be larger for old than
for new plants. All new plants can choose their location and, therefore,
at the outset face common labor markets. In fact, the dispersion in
average wages was larger for new than for old plants. This reflects the
role of competing technologies with substantial trade-offs between human
capital and other inputs rather than unionization or regional variations
in wages.

As a final check, we divided all plants into the nine census
geographic regions to assess to what extent variations in average wages
were attributable to regional influences on wage rates. We found that,
generally, one could not predict the regional pattern of high and low
wages for one industry from the observed pattern for another industry.

Output

The dependent variable, output, was proxied alternatively by
deflated shipments and by deflated value added.(3) Both shipments and
value added have deficiencies as measures of output. While shipments
ignore variations in purchases from other plants, value added is subject
to error in measuring cost of materials and to error from
inconsistencies in the valuation of inventories over time. The choice
between them as to which is the best measure of changes in output
depends partly on the set of industries examined. For this reason,
results are generally presented with both variables as alternatives.

Conceptually, when shipments are used as the proxy for output, one
might argue that materials inputs should appear on the right side of the
equation. However, materials inputs are so large a fraction of shipments
and, in consequence, so highly correlated with the latter at least in
the context of cross-section data, that the introduction of this
variable tends to dominate and obscure other relationships.

III. Estimates of Embodied Technical Change Based on Data for
New Plants

From the standpoint of our analysis, the central distinction
between "new" and "old" plants is not their
chronological age but whether, at the starting date for analysis, there
are initial endowments of physical capital that originate from earlier
investments. If there are none, or if they are minor, we classify the
plants as "new." The relevance of this distinction rests in
our hypothesis (developed more fully later) about interactions between
new investment and the initial capital stock. These interactions lead to
non-separability of the relation between new inputs and changes in
output and, hence, to unstable and misleading coefficients in the
context of a production function.

Interactions across successive investment streams also occur for
new plants with no initial endowments of capital. But, as explained
later, for new, in contrast to old, plants interaction effects are far
more likely to be proportional to cumulated new investment. First,
across new plants, the stock of capital is far more homogeneous in its
age composition. Second, for new plants investment streams of contiguous years are frequently elements of an integrated investment plan.

New plants were defined as plants born in 1973 or later while old
plants were those in existence in 1972 (the initial year for the
available data set). New plants were in fact considerably younger than
old plants and, further, the bulk of their capital outlays were made
within several years of their birth.

The Econometric Model

The production function, equation (4), is now expressed in
Cobb-Douglas form as equation (5):

O = [Ae.sup.at] [L.sup.[beta].sub.1.Q.sup.[beta].sub.2]
[(K.sub.v.e.sup.kv).sup.[beta].sub.3] (5) The customary Cobb-Douglas
specification was modified by the inclusion of an index of human capital
as a separate argument in the production function. Taking logs,
simplifying, substituting [[beta].sub.o] for log ([Ae.sup.a[tau]]) and
noting that [[beta].sub.4] = [[beta].sub.3]k, we have the empirical
specification in equation (6):

(6) where the variables O,L,Q,K and V are defined as before and
each variable is measured for plant j in time t. Note that the average
vintage variable appears in linear rather than log form as a direct
result of the specification in equation (5).

Results

Table I presents the results for equation (6) using the two
alternative proxies for output, shipments and value added. For the
analysis of new plants, two sets of industries were selected: one
comprising 41 manufacturing industries and a subset of 32 industries.
For the larger set, we included all industries with at least 16 new
plants in 1982 (excepting only NEC industries and several that might not
be considered primarily in manufacturing, e.g., publishing). For the
subset of 32, the cutoff was 20 new plants.(4)

The terminal peak for each plant was defined as the year with the
highest value of shipments in the period 1984-86. The cross-sections are
estimated with plant data for the peak year only. The intention behind
utilizing a "terminal peak" was to minimize measurement errors
associated with variations across plants in capacity utilization.

In addition, for each alternative the results are shown for all
plants born between 1973 and 1986 and then narrowed to those born at
least three years before the terminal peak for each plant. Limiting the
data to plants born at least three years prior to the terminal peak had
the purpose of allowing sufficient time for capital goods to be fully
phased in and, hence, for the estimates to correspond to die production
frontier.

Turning to the results, the coefficients for labor, human capital,
and physical capital are highly stable across the eight set of
estimates, the median values for the three coefficients being .64, .61,
and .33, respectively. There is considerably more volatility from sample
to sample for the coefficients for vintage, with the median value being
.04. The [R.sup.2] and t values, considering that cross-section data for
a highly diverse group of industries were used, are all very high.

The principal contributions of the econometric model is that it
enables the measurement of the effects on output of embodied technical
change, and of human capital separate from "pure labor." To
focus first on the latter, we observe from Table I that, particularly
for estimates with value added as the dependent variable, the elasticity
of output with respect to human capital is roughly the same as that for
pure labor.

Given our definitions, a one percent change in human capital
(measured by the average wage) must have the same effect on total costs
as a one percent change in the labor input (measured by number of
employees). Accordingly, the same coefficients for the two variables
mean that the marginal products per dollar of expenditures are the same
for the two inputs. The consistency of this result with an optimal input
allocation rule is an outcome one might have expected from data drawn
from an industry with a homogeneous output. It is surprising given the
variety of industries and technologies from which the plants were drawn,
as well as the enormous range of plant sizes encompassed by the samples.

[TABULAR DATA OMITTED]

The most frequent estimate for vintage yielded a coefficient that
indicated a four percent change in output for each one-year change in
the average vintage of the stock of capital. This is indeed a high value
given that gross returns to capital have a weight of roughly one-third
in total inputs for manufacturing industries (as measured by
capital's share of gross compensation to capital plus labor, and
using Statistics of Income data for 1972-86). Note also the consistency
of the share of capital with the .33 median elasticity in the
Cobb-Douglas model. Thus a 4 percent change in output attributed solely
to embodied technical change for physical capital implies about a twelve
percent change in the efficiency of capital goods from a one-year change
in average vintage (k = [[beta].sub.4]/[[beta].sub.3] = .04/.33).

While there was some instability in the estimated coefficients for
vintage, such estimates can only be viewed as rough approximations of
average rates of change attributable to the age of capital goods. Not
only is each year's investment composed of large numbers of
specific capital goods, but the functional composition of capital goods
(for example, structures versus equipment or office equipment versus
transportation equipment) undoubtedly changes across vintages. Thus, the
derived measures are meaningful only as approximations or scalar magnitudes rather than as point estimates. Our cross-sectional estimates
of embodied technical change are considerably higher than estimates by
Hulten [4] derived from ratios of quality adjusted to unadjusted
equipment prices.

A dummy variable model was specified to assess the differences
between estimated coefficients for each industry and those for the
aggregate sample. Very few of the industry dummies were significant -
that is, industry estimates for the coefficients did not deviate
significantly from the estimates for the combined sample. Thus, the
aggregate results in Table I are justifiable.

IV. Modeling Production Relations for Changes in Inputs and
Output

The average proportion of capital expenditures in U.S.
manufacturing that is spent on existing plants, as distinct from those
under construction, has been estimated by Gort and Boddy [2] to exceed
90 percent of the total. At first glance, this fact seems puzzling. The
addition of new capital goods to a production process already in place
and incorporating old assets must be restricted in the kinds and
combinations of inputs that can efficiently be added. Why then do firms
choose to give up the flexibility and consequent economies associated
with new plants of best practice technology?

There are three plausible explanations for investment in old
establishments. First, their expansion may entail a shorter gestation
period than creation of new plants. Second, scale economies may preclude the creation of new plants for small increases in output. And third,
total input requirements for a given increase in output may be smaller
for old plants because of interactions between old and new inputs. It is
this third explanation that is the focus of our attention.

There are two principal ways in which such interactions may occur.
First, new employees may learn from older ones thereby reducing
adjustment costs. Second, new physical assets may interact with old ones
by modifying them, or at least changing the way in which old assets are
used. In this way, new capacity could be created with lower inputs of
physical capital than required when starting from a zero base.

It is our hypothesis that increments in output entail different
production function coefficients from those implicit in levels of output
and inputs and that this, in turn, is a consequence of interactions.
Were it otherwise - that is, in the absence of interactions - new inputs
of capital (investment) on old plants could be viewed as separable levels of capital just as increases in output could similarly be viewed
as the level of new output.

Gort and Boddy [2] modeled interaction effects through a simple
multiplicative term - a procedure that made sense for the electric power
industry they studied since the interaction took largely the form of
addition of generating equipment to old structures, or of modifications
of boilers for existing steam turbogenerators. The assumption of a
symmetrical effect of new investment across all old capital goods as
implied by a multiplicative term is, however, much too simple to capture
the technological interactions observable in most industries. Indeed,
interactions are difficult to model since they are likely to vary across
plants within an industry as well as across industries.

Interactions occur across all vintages of investment. The fact that
one year's investment may be composed of structures while the next
year's is composed of equipment housed by the structures, means
that growth in output cannot be expected to respond in a consistent way
to a single year's investment outlays. Outlays over several years
are likelier to reflect a balanced investment plan than those for a
single year and, hence, (holding technology constant) are likelier to
produce a proportional relation between growth in output and cumulated
investment. But a balanced investment program still does not dispose of
interactions that take the form of modifications of old assets made
possible by new technology. Nor does it take account of differences in
returns to new investment from interactions arising because of large
differences across plants, at any point in time, in the size of the
initial stock of capital.

It is plausible, however, that interaction effects associated with
the stock of old assets existing at the outset decline as a function of
time, relative to the separable output effects of new investment. Old
plants vary not only in the magnitude of their initial capital
endowments but also in the age of their old capital. Consequently when
new investment is still small relative to old investment, interactions
between new and old capital will produce unsystematic and, hence,
unpredictable effects in the context of cross-section analysis. The
problem, therefore, reduces itself to one of finding a subsequent point
in time at which interaction effects across plants become sufficiently
systematic that they can be measured.

Consider equation (7) for old plants

[Mathematical Expression Omitted] (7) where [tau] is the vintage
year (with [tau] = O the base year), [O.sub.[tau]] is defined as output,
[Mathematical Expression Omitted] the vector of current vintage year and
previous investments such that [Mathematical Expression Omitted]
[K.sub.[omicron]] the initial capital stock, [L.sub.[tau]] is labor, and
[Q.sub.[tau]] is labor quality. The hypotheses concerning embodied
technical change can be summarized as follows:

[Mathematical Expression Omitted]

Equation (8) (i) shows, as before, the greater productivity of more
recent vintages of investment. The effect of interactions is shown in
equation (8) (ii) by the positive (if any) contribution of more recent
vintage investment, denoted by [tau], to the marginal productivity of
past investments denoted by [tau] - 1 and [tau] - j. Interaction effects
will be larger between investments of shorter time lapse between
vintages (i < j) because old assets become progressively less
adaptable to new capital. Finally, with obsolescence, the productivity
of the initial capital stock, equation (8) (iii), declines over time.
While not reflected in the above equations, the relative effect on
output of interactions with [K.sub.[omicron]] declines over time for
still another reason. As the sum of new investments grows over time,
their separable effects on output, and the interactions across the new
investments, grow in importance relative to the effects of
[K.sub.[omicron]].

If one assumes that both vintage and interaction effects are of no
consequence for investment (our null hypothesis), the production
function for vintage year [tau] can be expressed as

[O.sub.[tau]] =
[h.sub.[tau]]([K.sub.[tau]],[L.sub.tau]],[Q.sub.tau]] (9) where
[K.sub.[tau]] is the capital stock aggregated from the investment vector
and the initial stock. Now define [delta difference][O.sub[tau]] =
[O.sub.[tau]] - [O.sub.[omicron]], where [O.sub.[omicron]] =
[h.sub.[omicron]]([K.sub.[omicron]],[L.sub.[omicron]],[Q.sub.[omicron]])
under the null hypothesis stated above (that is, with no embodiment).
The increment to output relative to the base year level should then be a
separable production function(5) expressible as

The role of human capital in equation (10) is similar to that of a
technology index. One would expect that growth in physical output
depends not only on growth in physical inputs, but also on the
functional relationship between the existing level of human knowledge
and increments to physical capital and labor. Thus, [Q.sub.[tau]], is
the level of human capital reached by vintage year [tau] which can be
utilized by the increment to the labor force in the production of
additions to output.

Equation (10), if it holds, implies that the coefficients of the
production function (ignoring economies of scale) are the same whether
one estimates the relation for increments to output or for levels of
output (equation (9)).

Vintage effects, when included in an empirical specification,
permit a test to determine whether productivity is greater for more
recent additions to the capital stock. If interactions initially have an
unsystematic effect, this obscures the production relation of changes in
physical capital to increments in output for vintage years soon after
the base year (that is, the start of the period examined). However, as
the time elapsed from the base year increases, the production relation
for increments to output approaches that for a specification in terms of
levels rather than changes. A change in capital variable can thus be
assumed to capture the "levels" effect of a balanced
investment plan and, in addition, the systematic component of the impact
of interactions on productivity.

V. Estimates of Embodiment for Old Plants

Using cross-section data, we again estimate a modified Cobb-Douglas
production function, this time for changes in output and in labor and
capital inputs for old plants. The model is written as follows:

[Mathematical Expression Omitted]

where [delta difference] log [O.sub.jt] is the percentage change in
output for the jth plant for time t relative to the initial period of
1972, [delta difference] log L is the percentage change in pure labor,
and [delta difference]A log K is the percentage change in gross capital.
Percentage changes standardize units to control for size effects across
plants, and, in a sense, also standardize the observations for
differences in initial factor proportions.

Log Q measures the level of human capital available to the
increment in the labor force, and V is, as before, a measure of the
weighted average vintage of investment expenditures for each plant.
Output is again measured by the (deflated) value of shipments or,
altematively, value added, labor by total employees, and labor quality
or human capital by the average wage rate for each plant. The weights
for vintage are, of course, the annual investment expenditures for the
period over which changes in capital inputs are measured.

For each regression, the initial capital stock for each plant is
simply its deflated gross assets in 1972,(6) and the terminal capital
stock is obtained by adding to the initial value cumulated (deflated)
gross capital expenditures plus the capitalized value of the change in
rentals of assets. Errors associated with the measurement of initial
stocks, for which data on annual investment streams are lacking, can be
expected to reduce greatly the goodness of fit of our model.

Our objective was to test the implications of equation (8) that the
power of interactions gradually declines over time. Equation (11) was
therefore estimated consecutively for each year. According to the null
hypothesis of no embodiment in the form of vintage or interaction
effects, [[beta].sub.4t] = O and [[beta].sub.3t] > O, respectively,
in the years immediately following 1972. A positive and stable measured
effect for the [delta difference] log K variable would indicate
relatively weak interaction effects, and, hence, a separable production
relation for changes in output and capital input.

We now turn to results for the empirical model in equation (11),
for plants in 15 industries.(7) Table II is presented with shipments as
the proxy for output. The same estimates but with value added as the
dependent variable yielded very similar, though somewhat more erratic,
results with lower values of [R.sup.2]. For economy of space, the latter
are not reported in detail.

The results show the consecutive changes in coefficients for old
plants for the increments in output and inputs from 1972 to the levels
for each successive year. In general, there is strong support for the
conclusion that for an extended period, interactions with the initial
capital stock do not permit estimation of a separable relation between
change in capital inputs and the change in output. It takes roughly
twelve years for the relative effect of interactions to decline to a
level that permits one to estimate a stable coefficient for the change
in the log of K.

Over time, the coefficient for [delta difference] log K increases
from near zero and insignificance in the early 1970s to a significant
positive elasticity of above 0.3 by 1984. Initially, a systematic
relation between growth in capital and in output is obscured by the
unpredictable effects of interactions, given large variations across
plants in the age and size of the endowments of capital at the outset of
the period. But as the ratio of cumulated investment to initial capital
rises, interaction effects become more systematic and the coefficient
for [delta difference] log K measureable.

Technical change embodied in capital is shown most directly by V.
The insignificant results for V in earlier years were to be expected. If
the effects of increments in capital are obscured by interactions, it is
likely that so will the effects of changes in the vintage of capital.
Moreover, since it is the vintage of post-1972 capital that was
measured, sufficient time had to elapse for there to be enough
dispersion in vintage to detect an effect.

Accordingly, the coefficient of V did not become significantly
positive until 1982, but remained reasonably stable thereafter averaging
.04 for the five year interval 1982-86. The approximately 4 percent
increase in output for every one year change in vintage is substantially
the same as that observed earlier for new plants. Thus the high rate of
embodied technical change observed for new plants is confirmed with data
for old plants.

The consistently rising negative value of the intercept as one
moves from 1973 to 1986 is explained by the construction of the capital
variable. While V measured changes in the vintage of post-1972 capital,
no allowance was made for the progressive obsolescence of the initial
(1972) stock of capital. Hence, for all plants, terminal year capital
was systematically overstated by an increasing amount for each
successive year. Thus the rising negative intercept appears to capture
the obsolescence rate for old capital.

Table II gives us some insight into the effect of interactions
between new and old inputs of physical capital. It is problematic,
however, insofar as year-to-year changes may represent observations for
less than capacity utilization and, hence, may not correctly measure the
production frontier. This is especially a problem for the labor input
and may explain the instability of the coefficient for log of Q in Table
II. As is well known, firms retain skilled labor during contractions in
output. The resulting change in the composition of labor, with its
consequent change in the average wage, is likely to lead to some
distortion in the coefficients for L and Q. An illustration of this
phenomenon is reflected in the non-significant results for Q during the
1981-82 recession.

[TABULAR DATA OMITTED]

VI. Comparing Production Relations for New and Old Plants

The next objective was to derive estimates for all the variables in
equation (11) that correspond to the production frontier and this
required that we measure changes in output and inputs between points
approximating capacity utilization. Production relations involving
changes, as distinct from levels, of output and inputs are likely to be
especially sensitive to the assumption of capacity utilization and that
condition seems best approximated at output peaks.

For an empirical approximation of capacity utilization, the first
peak was the higher of the 1972 or 1973 value of shipments. The terminal
peak again was the year with the highest value of shipments in the
period 1984-86. The cross-section data are based on the peak to peak
change in logs for O, L, and K for each plant. In the overwhelming
majority of cases, the highest rate of capacity utilization by almost
any criterion did occur within those intervals.

The resulting estimates, with both shipments and value added as
proxies for output, were also compared with the coefficients for new
plants. The latter were derived from equation (6) but limited now to the
same 15 industry sample as used for old plants. The estimates are shown
in Table III.

[TABULAR DATA OMITTED]

Before proceeding with the comparison of old and new plants, some
characteristics of the results for old plants might be noted. As
compared with the average coefficients reported in Table II for
consecutive years, the coefficient for labor declines markedly. Those
for physical capital and vintage rise though the order of magnitude remains roughly the same as before. In estimating an industry dummy
variable model as for new plants in section III, most of the industry
dummies proved non-significant thereby rendering the coefficients for
the aggregate more meaningful as average estimates.

Comparing the results in Table III we find:

(a) As to be expected, the [R.sup.2] values for new plants are much
higher than for old plants - a fact attributable in large part to far
superior data for the capital variable for new plants.

(b) For new plants, the coefficient for human capital (Q) is
substantially higher than for old plants. The efficiency with which new
plants use human capital appears to be their single most important
advantage over old plants. Once technological options are limited by a
large amount of old physical assets, the ability to substitute human
capital for other inputs appears to be severely restricted.

(c) The coefficient for labor is of roughly the same magnitude for
old and new plants.

(d) Especially important is the considerable stability in the
coefficients for K when estimates based on changes in inputs and output
for old plants are compared with levels for new plants. However, while
these coefficients are generally of the same order of magnitude, the
coefficients remain slightly higher for old plants. This may suggest
some continued impact of interactions even after a period of as much as
14 years past the point at which the initial capital stock was measured.

(e) V continues to be much more sensitive than K to choice of
sample and proxy for output. However, the higher estimates for old than
for new plants are consistent with what we know about capital
expenditures. A larger proportion of capital outlays are devoted to
structures rather than to equipment for new than for old plants.
Structures are generally assumed to be associated with much lower rates
of obsolescence (hence, embodied technical change).

VII. Summary

There is renewed interest in the recent literature in measurement
and measurement errors associated with output growth. The standard
approach, such as in Hulten[4], is to compute Solow residuals by
decomposing output growth into the contributions of input quantities,
quality adjustments, and the relative proportions of embodied and
disembodied technical change. This paper provides an alternative
approach to measuring the vintage effects of additions to capital
utilizing cross-sections of manufacturing industries at the plant level
made possible by the Census LRD database.

The principal results of the paper are briefly summarized:

1. We first specified a production function with human capital as a
separate argument and with embodied technical change proxied by a
variable that measures the average vintage of the stock of capital.

2. The coefficients of the production function were first estimated
with cross section data for roughly 2150 new manufacturing plants in 41
industries, and for subsets of this sample. An augmented Cobb-Douglas
specification was used. The results proved fairly stable across varying
samples of plants and with respect to alternative measures of output.

3. Substantively, it was found that the elasticity of output with
respect to human capital was approximately the same as it was with
respect to pure labor. Embodied technical change of capital produced an
average 4 percent increase in output for each one year change in average
vintage.

4. It was pointed out that most investment in a developed economy
is made on old rather than on new plants. An important question,
therefore, concerns the separability of the relation between new inputs
(that is, changes in the level of inputs) and changes in the level of
output. A model was specified with interactions between new investment
and initial endowments of capital. Interaction effects were predicted to
decline in importance as a function of time.

5. Using a sample of roughly 1400 old plants in 15 industries, it
was found that interactions between new investment and initial
endowments of capital were, for a long interval of time, too
unsystematic to permit measurement of a coefficient for capital. After
twelve to fourteen years of cumulative investment, a systematic relation
between changes in the level of inputs and changes in the level of
output became measurable. Moreover, the coefficient for changes in the
capital input for old plants proved to be of approximately the same
magnitude as that for level of capital for new plants.

6. Comparing new and old plants over the "long-run," the
estimates of embodied technical change of capital and of the elasticity
of output with respect to number of employees (pure labor) proved very
similar for the two types of plants. Differences between new and old
plants in the elasticity of output with respect to human capital
remained very large, however, and appear to be an important difference
between the two sets of plants.

References

[1.] Becker, Gary. Human Capital. The University of Chicago Press,
1964. [2.] Gort, Michael, and Raford Boddy. "Vintage Effects and
the Time Path of Investment in Production Relations," in The Theory
and Empirical Analysis of Production, Studies in Inconte and Wealth,
Vol. 31, National Bureau of Economic Research. New York: Columbia
University Press, 1967, pp. 395-422. [3.] Hall, Robert E.,
"Technical Change and Capital from the Point of View of the
Dual." Review of Economic Studies, January 1968, 35-46. [4.]
Hulten, Charles R., "Growth Accounting When Technical Change is
Embodied in Capital." American Economic Review, September 1992,
964-80. [5.] Prucha, Ingmar R., and M. Ishaq Nadiri. "Endogenous
Capital Utilization and Productivity Measurement in Dynamic Factor
Demand Models: Theory and an Application to the U.S. Electrical
Machinery Industry." Working paper, 1990. [6.] Solow, Robert M.,
"Technical Change and the Aggregate Production Function."
Review of Economics and Statistics, August 1957, 312-20.

(*) The project was carried out with assistance from the
ASA/NSF/CENSUS fellowship program. The authors are, of course, solely
responsible for the conclusions and methods of analysis used. (1.)
Investments were deflated by the implicit price deflator for capital
expenditures in all manufacturing combined, based on unpublished Bureau
of Economic Analysis data. To the cumulative total of gross capital
expenditures we added the capitalized value of the changes in rentals of
fixed assets. The rate of capitalization was derived from the ratio of
gross fixed assets to the sum of net income before taxes plus interest
paid plus depreciation, as reported for 1972-86 in U.S. Internal Revenue
Service, Statistics of Income. (2.) To facilitate the interpretation of
the coefficients of change in human capital, as well as to correct for
possible biases arising from the fact that in some of our cross-sections
the observations do not relate to identical points in time, average wage
rates were deflated. The deflator was the Consumer Price Index and was
intended simply to correct for the average rate of inflation in the
economy. (3.) Output deflators were also drawn from unpublished BEA data
at the 4-digit industry level. (4.) Within the sets of industries for
both new and old plants, only plants that satisfied the following
criteria were chosen: (a) a continuous history in the same industry,
from birth for new plants and from 1972 for old, until 1986, (b) a
primary industry specialization ratio of at least 50%. This gave us
about 2150 new plants for the 41 industries and roughly 1900 for the 32.
The period chosen, 1972-86, was determined by the time interval for
which panel data were available. A list of industries and number of
plants in each will be provided upon request. (5.) The mathematical
conditions necessary for exact separability of output and input levels
into functions based on increments is, to our knowledge, unsolved. (6.)
The deflators were derived from ratios of gross capital stocks at
historical cost to stocks at constant cost as reported in Bureau of
Economic Analysis, U.S. Department of Commerce, Fixed Reproducible
Tangible Wealth in the United States, 1925-85, 1987. (7.) For the
analysis of old plants, and for comparisons of old and new plants, a
further subset of 15 industries was employed primarily to keep the size
of the database manageable. These consisted generally of the largest
among the original 41 industries but with selection based on broad
representation across the industrial spectrum. Our sample consisted of
about 1400 old plants in the 15 industries, and about 1250 new plants.

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